%!TEX program = xelatex
\documentclass[t,12pt,aspectratio=169]{beamer} % 16:9 宽屏比例
\usepackage{ctex} % 中文支持
\usepackage{amsmath, amssymb, bm}
\usepackage{graphicx, color}
%\usepackage{natbib}
\usepackage{hyperref}
\usepackage{booktabs}

\usepackage{tikz} % 可选：用于绘图
\usetikzlibrary{shapes,arrows,arrows.meta}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 或者更彻底地使用 Latin Modern 数学字体（需 lualatex/xelatex）
\usepackage{unicode-math}
%\setmainfont{Latin Modern Roman}
\setmathfont{Latin Modern Math}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{tikz} % 用于绘图
\usetikzlibrary{shapes,arrows}
\usetikzlibrary{circuits.ee.IEC} % 画电路
%\usetikzlibrary{arrows.meta, calc, positioning, quotes, angles, 3d}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 画函数图形
\usepackage{pgfplots} 
\pgfplotsset{compat=1.18} % 推荐设置兼容性版本

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{setspace}
\onehalfspacing
\setlength{\parskip}{1em}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\addtobeamertemplate{frametitle}{}{\vspace*{0.3em}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usetheme{Madrid}
\usecolortheme{default}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{追击曲线问题}
\author{五六七}
%\date{2025年9月15日}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}

% 封面页
\begin{frame}
  \titlepage
\end{frame}

% 目录页
\begin{frame}{目录}
  \tableofcontents
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{追击曲线问题 -- 建立常微分方程}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[allowframebreaks]{追线问题 -- 建立常微分方程}

%（Pursuit Curve）

\vspace{-0.3cm}

问题：设某人从原点出发，沿着$x$轴正向跑步。速度大小恒定为$v_1$. 
设小狗从位置$B(0,1)$出发，始终朝向此人奔跑，速度大小恒定为$v_2$.
求小狗的运动轨迹所满足的常微分方程。

\begin{figure}[h]
    \centering
    \begin{tikzpicture}[scale=2.0, >=Stealth, thick]

        % 定义点的位置
        \coordinate (A) at (0,0);
        \coordinate (B) at (0,1);
        
        \coordinate (A1) at (0.6,0);
        \coordinate (B1) at (0.25,0.6);

        \coordinate (A2) at (1.2,0);
        \coordinate (B2) at (0.7,0.4);

        % 绘制x轴、y轴
        \draw[->, -{Stealth[scale=1.5]}, thick] (-0.5,0) -- (3,0) node[below right] {$x$};
        \draw[->, -{Stealth[scale=1.5]}, thick] (0,-0.5) -- (0,1.5) node[below right] {$y$};

        % 绘制点
        \fill[blue] (A) circle (1.5pt) node[below left] {$A$};
        \fill[purple] (B) circle (1.5pt) node[left] {$B$};
        \draw[->, -{Stealth[scale=1.3]}, thick, purple] (B) --++ (0,-0.36);
                
        \fill[blue] (A1) circle (1.5pt) node[below] {$A_1$};
        \fill[purple] (B1) circle (1.5pt) node[above] {$B_1$};
        \draw[->, -{Stealth[scale=1.3]}, thick, purple] (B1) --++ (0.18,-0.31);

        \fill[blue] (A2) circle (1.5pt) node[below] {$A_2$};
        \fill[purple] (B2) circle (1.5pt) node[above] {$B_2$};
        \draw[->, -{Stealth[scale=1.3]}, thick, purple] (B2) --++ (0.27,-0.22);

      \end{tikzpicture}
%    \caption{追击曲线}
%    \label{fig:pursuit-curve}
\end{figure}





\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{方程求解}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[allowframebreaks]{方程求解}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{作图}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[allowframebreaks]{作图}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{参考文献}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{参考文献}

\begin{thebibliography}{99}

\bibitem{weisstein} Eric W. Weisstein. Pursuit Curve. From MathWorld -- A Wolfram Resource. 
\url{https://mathworld.wolfram.com/PursuitCurve.html}

\bibitem{chash} O. I. Chashchina, Z. K. Silagadze. The dog-and-rabbit chase problem as an exercise in introductory kinematics. 
\url{https://arxiv.org/abs/0711.3293v2}

\bibitem{lloyd} Michael Lloyd. Pursuit Curves. Lecture Notes. 

\bibitem{bernhart} Arthur Bernhart. Curves of General Pursuit, Scripta Mathematica, 24, 1959, pp. 180–206. 

\end{thebibliography}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}

